本研究目的在於對上市公司的隱含違約機率進行估計，進而預測公司違約可能性。我們利用混合對數模型來對厚尾機率(違約部分)進行捕捉，故可較為精確得估計隱含違約機率。為達此研究目的，本研究以OptionMetrics資料庫的31家破產公司為研究對象，研究各公司破產前三個月內的每日違約機率。我們採用最小平方法，將市場之選擇權價格與本研究推導出的選擇權理論價格，配適出模型參數，進而估計隱含違約機率。

為了解模型所獲得之隱含違約機率是否準確，我們不只研究破產公司的違約機率，亦挑選至今仍健在的公司作為對照組，以檢驗模型的預測能力是否精準。我們另以CAP方法，將研究數據與CDS所得出之違約機率做比較。另外，將兩種模型各自對違約公司與對照組做T檢定，以此檢驗此兩種模型對違約是否有顯著區分能力。

關鍵字：價格密度函數、機率預測、選擇權市場、CDS、模型配適

The purpose of this study is to estimate the implied default probabilities for listed companies. And then we can forecast the possibility of companies default. We use the mixed lognormal model to capture the fat tail of probability (default part), so we can estimate the implied default probabilities more accurately. In this study, 31 bankrupt companies in OptionMetrics database were selected, and the daily probability of default was estimated by three months before the company went bankrupt.

We use the least squared pricing errors method to estimate the model parameters, and then compute the implied default probabilities. In order to understand the accuracy of the implied probability of default obtained by this research model, we not only study the default probability of the bankrupt company, but also select a few companies that are still strong to determine whether the prediction ability of the model is great. We also compared our model to CDs by CAP. Besides, We do T test -for our model and CDS estimation method separately to understand whether they can distinguish default companies from the non-default counterparts.

Key word: pricing density function, probability prediction, option market, CDS, model matching

Altman, E., (1968) Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy. Journal of Finance, 23, 589-609.

Bao, Y., Lee, T., Saltoglu, B., (2007). Comparing density forecast models. Journal of Forecasting, 26, 203–225.

Beaver, W., (1966) Financial Ratios as Predictors of Failure. Supplement to Journal of Accounting Research, 4, 71-111.

Bharath, S. T., and Shumway, Tyler., (2008). Forecasting Default with the Merton Distance to Default Model. The Review of Financial Studies, 21, 1339-1369.

Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economics,81(3), 637-654

Black, F. and Cox, J., (1976) Valuing Corporate Securities: Some Effects of Bond Indenture Provisions. Journal of Finance, 31, 351-367.

Câmara, A., I. Popova and B. Simkins, (2012). A Comparative Study of the Probability of Default for Global Financial Firms. Journal of Banking and Finance, 36 , 717-732.

Chang, B. Y., and Orosi, G., (2016). Equity Option Implied Probability of Default and Equity Recovery Rate. Journal of Futures Markets, 10.1002

Chen, Z., and W. Fong. (2012). Was the Writing on the Wall? An Options Analysis of the 2008 Lehman Brothers Crisis. Journal of Investment Management, 10, 1-12.

Christoffersen, P., and Mazzotta, S. (2005). The accuracy of density forecasts from foreign exchange options. Journal of Finanical Econometrics, 3, 578–605.

Cochrane, J. H., (2001). Asset Pricing. Chicago: Princeton.

Diebold, F., Gunther, T., and Tay, A. (1998). Evaluating density forecasts with applications to financial risk. International Economic Review, 39, 863–883.

Diebold, F., Hahn, J., and Tay, A. (1999). Multivariate density forecast evaluation and calibration in financial risk management: high-frequency returns on foreign exchange. Review of Economics and Statistics, 81, 661–673.

Engle, R. F. and Manganelli, S. (2004). Caviar: Conditional Autoregressive Value at Risk by Regression Quantiles. Journal of Business and Economic Statistics, 22 (4), 367-381.

Fan, R., Taylor, S. J. and Sandri, M. (2017). Density forecast comparisons for stock prices, obtained from high‐frequency returns and daily option prices. Journal of Futures Markets, 38(1), 83-103.

Giacomini, R. and Komujer, I. (2005). Evaluation and Combination of Conditional Quantile Forecasts. Journal of Business and Economics Statistics, 23(4), 416-431.

Hillegeist, S. A., Keating, E. K., Cram, D. P., and Lundstedt, K. G. (2004). Assessing the probability of bankruptcy. Review of Accounting Studies, 9, 5-24.

Hua, J. and Manzan S. (2013). Forecasting the return distribution using high-frequency volatility measures. Journal of Banking and Finance, 37(11), 4381-4403.

Koenker, R. and Bassett, G. (1978). Regression Quantiles. Econometrica, 46, 33-50.

Kris B., Mikael P., (2014). Intraday liquidity dynamics and news releases around price jumps: Evidence from the DJIA stocks. Journal of Financial Markets, 17, 121–149.

Liu, X., Shackleton, M. B., Taylor, S. J., and Xu, X. (2007). Closed-form transformations from risk-neutral to real-world densities. Journal of Banking and Finance, 31(5), 1501-1520.

Merton, R., (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 29, 449-470.

Merton, R.C., (1976) Option pricing when underlying stock returns are discontinuous. Journal of Finance Economics, 3, 125-144.

Petersen, M., and Rajan, R., (2002). Does Distance Still Matter? The Information Revolution in Small Business Lending. Journal of Finance, 57, 2533-2570.

Rainer, J., Florian, N., and Marti, G., Subrahmanyam. (2014). The determinants of recovery rates in the US corporate bond market. Journal of Financial Economics, 114, 155-177.

Rubinstein, M., (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics and Management Science, 7, 407-425.

Sarbu, S., Schmitt, C., and Marliese, U. H., (2013). Market expectations of recovery rates. SSRN Electronic Journal, 10.2139

Taylor, S. J., Tzeng, C. F., and Widdicks, M., (2014). Bankruptcy Probabilities Inferred from Option Prices. Journal of Derivatives, 22, 8-31.